Quantum dynamics of a driven parametric oscillator in a Kerr medium

In this paper, we first analyze a parametric oscillator with both mass and frequency time-dependent. We show that the evolution operator can be obtained from the evolution operator of another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation $t\rightarrow\int_0^t dt'\,1/m(t')$. Then we proceed by investigating the quantum dynamics of a parametric oscillator with unit mass and time-dependent frequency in a Kerr medium under the influence of a time-dependent force along the motion of the oscillator. The quantum dynamics of the time-dependent oscillator is analyzed from both analytical and numerical points of view in two main regimes: (i) small Kerr parameter $\chi$, and (ii) small confinement parameter $k$. In the following, to investigate the characteristics and statistical properties of the generated states, we calculate the autocorrelation function, the Mandel $Q$ parameter, and the Husimi $Q$-function.